The polynomial ax³ – 10x² + bx + 8, where a and b are constants, is denoted by p(x). It is given that (x – 2) is a factor of both p(x) and p'(x).

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

1. [M1] Substitute x = 2, equate to zero Or divide by x – 2 and equate constant remainder to zero.
2. [A1] Obtain a correct equation, e.g. 8a – 40 + 2b + 8 = 0 Seen or implied in subsequent work.
3. [M1] Differentiate p(x), substitute x = 2 and equate result to zero Or divide by x – 2 and equate constant remainder to zero.
4. [A1] Obtain 12a - 40 + b = 0, or equivalent SOI in subsequent work.
5. [A1] Obtain a = 3 and b = 4
6. Alternative method for question 5(a)
7. [M1] State or imply (x-2)² is a factor
8. [A1] p(x)=(x-2)² (ax+2)
9. [M1] Obtain an equation in b
10. [A1] e.g. by comparing coefficients of x: b=4a-8
11. [A1] Obtain a = 3 and b = 4
12. SC If uses x = -2 in both equations allow M1 and allow A1 for a = -3, b = -4.
13. [M1] Attempt division by (x – 2) The M1 is earned if division reaches a partial quotient of ax² + kx, or if inspection has an unknown factor ax²+ex+f and an equation in e and/or f. Where a has the value found in part 5(a).
14. [A1] Obtain quadratic factor 3x² – 4x – 4
15. [A1] Obtain factorisation (3x+2)(x-2)(x-2)
16. Alternative method for question 5(b)
17. [B1] State or imply (x-2)² is a factor
18. [M1] Attempt division by (x-2)², reaching a quotient ax + k or use inspection with unknown factor cx + d reaching a value for c or for d
19. [A1] Obtain factorisation (3x+2)(x-2)² Accept 3(x+2/3)(x-2)².